#5935×AGL(1,𝔽₁₇) #72254644

order	52224
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

#5935×AGL(1,𝔽₁₇)

Splittings

Fitting	C₁₇×H(5,𝔽₂) ⋊ D₃×C₁₆
derived	C₁₇×#997 ⋊ C₂×C₁₆
Sylow	C₁₇ ⋊ C₁₆×#5935

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#1020×AGL(1,𝔽₁₇)
commutator, hypocenter	C₁₇×#997	C₂×C₁₆
Fitting	C₁₇×H(5,𝔽₂)	D₃×C₁₆
socle	C₂×C₁₇	C₁₆×#1020
prime radical	D₁₇×#5935	C₈
cosocle	#997×#3633	C₂²
LCS	G ▷ C₁₇×#997
UCS	C₁ ◁ C₂

2348 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 2348 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 136

A minimal faithful permutation representation — 2 generators on 25 points:

(2,7,5)(3,8,6)(10,11,12,16,13,17,15,22,14,18,19,24,20,21,23,25);(1,5,7,2,4,8,3,6)(9,10,15,17,20,18,24,22,12,19,25,16,11,13,21,23,14)

Conjugacy classes & character table

221 conjugacy classes — view the full character table.

Relationships

holomorph of

Q₈×C₁₇

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,7,5)(3,8,6)(10,11,12,16,13,17,15,22,14,18,19,24,20,21,23,25), (1,5,7,2,4,8,3,6)(9,10,15,17,20,18,24,22,12,19,25,16,11,13,21,23,14));